A constraint-preserving neural network approach for mean-field games equilibria

Neural network-based methods have demonstrated effectiveness in solving high-dimensional Mean-Field Games (MFG) equilibria, yet ensuring mathematically consistent density-coupled evolution remains a major challenge. This paper proposes the NF-MKV Net, a neural network approach that integrates process-regularized normalizing flow (NF) with state-policy-connected time-series neural networks to solve MKV FBSDEs and their associated fixed-point formulations of MFG equilibria. The method first reformulates MFG equilibria as MKV FBSDEs, embedding density evolution into the equation coefficients within a probabilistic framework . Neural networks are then employed to approximate value functions and their gradients. To enforce volumetric invariance and temporal continuity , NF architectures impose loss constraints on each density transfer function. Theoretical analysis establishes the algorithm’s validity, while numerical experiments across various scenarios including traffic flow, crowd motion, and obstacle avoidance , demonstrate its capability in maintaining density consistency and temporal smoothness. • NF-MKV Net integrates NF and NNs for high-dimensional MFG solutions. • Process-regularized NF ensures density transformation consistency across time steps. • State-policy-connected time series neural networks approximates time-consistent solutions. • Validated on Traffic Flow, Crowd Motion, and Obstacle Tasks, distinguishing OT. • Outperforms existing methods with volumetric and time-continuity constraints.